Question: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{6p^3 - 30p^2 - 216p}{-7p^2 - 63p - 140}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {6p(p^2 - 5p - 36)} {-7(p^2 + 9p + 20)} $ $ y = -\dfrac{6p}{7} \cdot \dfrac{p^2 - 5p - 36}{p^2 + 9p + 20} $ Next factor the numerator and denominator. $ y = - \dfrac{6p}{7} \cdot \dfrac{(p + 4)(p - 9)}{(p + 4)(p + 5)}$ Assuming $p \neq -4$ , we can cancel the $p + 4$ $ y = - \dfrac{6p}{7} \cdot \dfrac{p - 9}{p + 5}$ Therefore: $ y = \dfrac{ -6p(p - 9)}{ 7(p + 5)}$, $p \neq -4$